Unified Geometrically Nonlinear Formulation of All Higher-order Shear Deformation Theories for Cross-ply Plates

Several higher-order shear deformation theories have been proposed for laminated plates, based on the expansions of displacements across the thickness, which are the same for all layers. In this study, a unified formulation of all higher-order theories is presented for cross-ply laminated plates based on polynomial expansions of displacements in the thickness coordinate z. It includes all the models available in literature. The governing equations for linear static and free vibfation response, and for buckling under inplane load are derived. The expressions for the stiffness matrix, inertia matrix, geometric stiffness matrix, and the load vector are developed for a simply supported rectangular plate using Navier's solution. A general purpose, single programme has been developed for all higher-order laminated plate theories

First order shear deformation theory for hybrid cylindrical panel in cylindrical bending considering electrothermomechanical coupling effects

The coupled first order shear deformation theory based on Flugge's approximations is presented for cylindrical bending of cross-ply laminated circular cylindrical hybrid panels without assuming a priori the distribution of electric potential across the thickness of the laminate. Unlike available two dimensional theories, the coupled constitutive equations for electric displacement are used in the charge equation of equilibrium to obtain its solution in terms of displacements of the panel. Fourier series solutions are obtained for simply-supported panels under static electrothermomechanical load. These are used in three dimensional equilibrium equations to obtain transverse stress components. The results are compared with the three dimensional solution to assess the accuracy of the present theory. The influence of the electrothermomechanical coupling on the response is found to be significant. The study reveals that it is important to incorporate what are called ‘thermal thickening’ and ‘electrical thinning/thickening’ effects in the two dimensional theory to improve its performance for cylindrical shell panels under thermoelectric loads

Assessment of third order smeared and zigzag theories for buckling and vibration of flat angle-ply hybrid piezoelectric panels

A recently developed improved third order theory (ITOT) for angle-ply hybrid piezoelectric plates in cylindrical bending is extended to include geometric non-linearity in the Von Karman sense. The transverse deflection is approximated non-uniformly to explicitly account for the transverse strain due to temperature and electric potential. The coupled non-linear equations of motion and the boundary conditions are derived using the extended Hamilton’s principle. The non-linear theory is used to obtain the buckling and free vibration response of symmetrically laminated hybrid angle-ply panels under inplane electro-thermomechanical loading. This theory and the third order zigzag theory with additional layerwise terms for inplane displacements are assessed in direct comparison with the exact 2D piezothermoelasticity solutions for forced harmonic response, buckling and free vibration response under initial inplane electro-thermomechanical loading. The comparison establishes the accuracy of the results of the zigzag theory and its superiority over the ITOT for the dynamic and buckling response of angle-ply hybrid panels

Coupled FSDT for piezothermoelectric hybrid rectangular plate

The first order shear deformation plate theory is presented for simply-supported, cross-ply laminated, rectangular hybrid plate without assuming a priori the distribution of electric potential and temperature across its thickness. The coupled constitutive equations for electric displacement are used in the charge equation of equilibrium to obtain its solution in terms of displacements of the plate. The Navier type solution for static thermoelectric load is used in three dimensional equilibrium equations to get transverse stress components. This theory is assessed by comparison with the three dimensional solution. The influence of the coupled effects is found to be significant for relatively thick piezoelectric layers

Assessment of a layerwise theory of hybrid beams for patch load

In this chapter, the accuracy of an efficient coupled one-dimensional zigzag theory for piezoelectric hybrid beams is assessed for composite and sandwich hybrid beams under static uniformly distributed pressure and electric potential loads on the whole beam or a part of it. The effect of various parameters on the accuracy is analyzed in the chapter

Three dimensional axisymmetric piezothermoelastic solution of clamped circular elastic plate bonded to piezoceramic plate

A three dimensional (3D) solution in terms of potential functions is presented for a finite clamped circular isotropic elastic plate bonded to a piezoceramic plate subjected to axisymmetric thermoelectromechanical load. The boundary and interface conditions are satisfied using Fourier-Bessel expansions yielding an infinite system of algebraic equations for the arbitrary constants. These are solved to any desired degree of accuracy by truncating to a finite set of equations. Results are presented to illustrate the effect of the thickness parameter. The present solution would help to ascertain the accuracy and range of validity of two dimensional (2D) theories of piezoelectric hybrid plates

Three-dimensional piezothermoelastic solution for shape control of cylindrical panel

A three-dimensional exact solution is presented for a simply supported, cross-ply laminated finite cylindrical panel with some piezoelectric layers subjected to thermoelectromechanical load. The variables are expanded in Fourier series to satisfy the boundary conditions at the ends. The solution is constructed as a product of an exponential function and a power series. The field equations yield characteristic equations for the exponents and recursive relations for the coefficients of the power series. Results are presented for thermoelectrical load to illustrate the effect of the thickness parameter. The feasibility of reducing deflection and stresses due to thermal load by actuating a piezoelectric layer is reported The effectiveness of various actuating schemes for shape control of thermally loaded panel is investigated

Exact piezothermoelastic solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending

This work presents an exact piezothermoelastic solution of infinitely long, simply supported, cylindrically orthotropic, piezoelectric, radially polarised, circular cylindrical shell panel in cylindrical bending under thermal and electrostatic excitation. The general solution of the governing differential equations is obtained by separation of variables. The displacements, electric potential and temperature are expanded in appropriate Fourier series in the circumferential coordinate to satisfy the boundary conditions at the simply-supported longitudinal edges. The governing equations reduce to Euler-Cauchy type of ordinary differential equations. Their general solution involves six constants for each Fourier component. These are solved from the algebraic equations obtained by satisfying the boundary conditions at the lateral surfaces. The solution of the inverse problem of inferring the applied temperature field from the given measured distribution of electrical potential difference between the lateral surfaces of the shell has also been presented. Numerical results are presented for typical thermal and electrostatic loadings for various values of radius to thickness ratio